Mini Chapter Seven
Delta and Gamma Hedging
Delta Hedging
As a concept means neutralising the sensitivity of an option’s price due to change in the price of the underlying stock. The option owner can buy or sell the underlying stock’s value equivalent to the delta value of the option. For example – being long one put option contract (one contract = 100 units of stock) that has a short delta of 0.5 (option loses 0.5% for a 1% move up in the underlying) can be neutralised by buying 50 shares at the current price. Thus the value of a 1% up move (down) in the stock price which reduces (increases) option price by 0.5% (owing to negative delta) would get compensated by the 1% appreciation (depreciation) of the 50 shares purchased to hedge the delta. Note that this combined position remains delta-neutral only for small changes in the underlying’s price. For larger changes, the delta of the option changes because of its underlying gamma which we will discuss next.
Gamma Hedging
To recap gamma denotes the change in delta of an option with a small change in the price of the underlying. So our delta neutral position created above does not remain neutral once the price of the underlying changes. This would mean re-adjusting the position that one had acquired in the underlying stock to neutralise the new delta. This constant adjustment on the back of changing hedge-delta is known as gamma hedging. A positive Gamma position would mean going longer delta on the option as the underlying rises and going shorter delta as the underlying falls. Delta hedging a positive gamma position would thus imply selling the underlying when the price rises and buying it when price falls. This should in itself yield positive returns which is then weighed against the time decay / theta of being long the option. Conversely negative gamma positions would mean buying high and selling low to hedge one’s delta and implies a cost to be paid for keeping the portfolio delta neutral.
Delta and Gamma Hedging
Consider a Long Call position ATM strike = 60
One option contract = One unit of stock.
Table 1 : Snapshot of Change in the underlying and the respective Delta and Gamma positions
| Stock Price ($) | Option Price ($) | Delta | Gamma |
|---|---|---|---|
| 54 | 2.5 | 0.3 | |
| 56 | 3.2 | 0.3 | 0.0 |
| 58 | 4.0 | 0.4 | 0.0 |
| 60 | 5.0 | 0.5 | 0.0 |
| 62 | 6.0 | 0.6 | 0.0 |
| 64 | 7.2 | 0.7 | 0.0 |
| 66 | 8.5 | 0.7 |
With the current underlying price at 60 we would sell 0.5 stocks of the underlying for every option purchased (for simplicity we would consider 100 option contracts and sell 50 stock) to create a delta neutral position to begin with. We shall follow a strategy of neutralising delta at the end of every day based on the delta of the option at the closing price of the stock on that day and look at two different scenarios.
- End of day (EoD) profit = Option profit + Delta hedge profit = 𝚫 option price x option contracts + 𝚫 stock price x Previous EoD delta position
Scenario One
- Day 1 – Stock moves up to 62, Day 2 – moves down to 58, Day 3 – moves back up to 60
- Day 1 profit = (1 x 100) + (2 x -50) = 0
- Day 1 EoD Delta Hedge = sell 10 more stock @ price of 62 to achieve a delta hedge of 0.6
- Day 2 profit = (-2 x 100) + (-4 x -60) = + $40
- Day 2 EoD Delta Hedge = buy 20 stock to be net short 40 at 58, to reach 0.4 delta at 58
- Day 3 profit = (1 x 100) + (2 x -40) = + $20
- Day 3 EoD Delta Hedge = sell 10 stock at 60 to be net short 50, to reach 0.5 delta at 60
- Total P&L over 3 days = $60 coming from hedging delta. Note that we are back to the same starting position as of Day 0.
Scenario Two
- Day 1 – Stock moves down to 56, Day 2 – moves up to 64, Day 3 – moves back down to 60
- Day 1 profit = (-1.8 x 100) + (-4 x -50) = + $20
- Day 1 EoD Delta Hedge = buy 16 more stock @ price of 56 to achieve a delta hedge of 0.34
- Day 2 profit = (4 x 100) + (8 x -34) = + $128
- Day 2 EoD Delta Hedge = sell 32 stock to be net short 66 at 64, to reach 0.66 delta at 64
- Day 3 profit = (-2.2 x 100) + (-4 x -66) = + $44
- Day 3 EoD Delta Hedge = buy 16 stock at 60 to be net short 50, to reach 0.5 delta at 60
- Total P&L over 3 days = $192 coming from hedging the delta. Note that we are back to the same starting position as of Day 0.
To summarise a move in any direction on the stock on a delta hedged option position would generate profits due to positive gamma in the position. Note we have ignored the Theta (time decay on long options) in the position so far. Hence the net daily profits should be adjusted for the time decay. This also implies there should be a breakeven move in the underlying everyday for it to compensate for the time decay. If one thinks of this breakeven move in terms of volatility then the realized volatility has to be equal to what was implied in the price of the option; $ breakeven amount (earned via positive gamma hedging) would be the same as dollars spent on premium/theta decay.
It’s important to note that as the realised volatility moves higher (as observed in scenario 2) delta hedging in a positive gamma backdrop will have superior returns. Also note that for double the move in the underlying’s price, profits on a delta hedged position would be more than double because of positive gamma kicking in. We obviously need to take into account theta loss on the long option over the 3 days but that theta loss is the same on scenario 1 vs 2 assuming Implied Vol hasn’t changed.
The hedging above has been explained for an option buyer. In the same vein one can think of the hedging (opposite side) for an option seller/writer. Since the writer would lose money on the hedges owing to the negative gamma in the position it is this cost that is importantly built into the option premium while quoting to the buyer.
